About defuzzification methods influence on fuzzy traveling salesman problem’s solving

Abstract

The article investigates the approach to using fuzzy numbers and the method of dynamic programming to find solutions to the traveling salesman problem, considering the fuzzy representation of time in real travel conditions. This allows for formulating a fuzzy optimization problem to find the best value of the objective function, which is determined by the amount of time required to travel between cities. The traveling salesman problem (TSP) is a classic problem of combinatorial optimization, which involves finding the shortest or fastest route among a set of cities. Fuzzy numbers are used to formalize the uncertainty and imprecision of input data, associated with the subjectivity in estimates of the duration of necessary travel intervals. For operating with fuzzy numbers, their transformation into a special form is proposed, and the formalization of the obtained fuzzy results into a crisp representation is carried out based on the center of gravity (CoG) method. A comparison of the results obtained based on solving the deterministic traveling salesman problem using defuzzified time distances and the defuzzification of the solution to the fuzzy traveling salesman problem was conducted. The results confirmed the dependency of the solution on the method of defuzzification. A program was developed that was used to compare the results of the traveling salesman problem using crisp and fuzzy numbers based on the dynamic method. A conclusion is drawn, indicating that the use of trapezoidal fuzzy numbers with the dynamic programming method leads to improved results of the problem compared to using crisp numbers based on the defuzzification of fuzzy distances. Methods of implementation and problematic areas of application of the computation results are presented and analyzed, demonstrating the constructiveness of the proposed approach for studying real processes.